Finite Group Databasebeta

← #13048725

Subgroups — #13048725

Order 1296.

576 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 77 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10619994C211yesmaximal, characteristic, complemented
2C2×C3×#1084C211yesmaximal, characteristic, complemented
3#67×H(3,𝔽3)C211yesmaximal, characteristic, complemented
4#10619894C211yesmaximal, characteristic, complemented
5#10619892C211yesmaximal, characteristic, complemented
6#10619890C211yesmaximal, characteristic, complemented
7D3×#1084C211yesmaximal, characteristic, complemented
8C3×#3804—34nomaximal
9#88744—31nomaximal
10C22×C3×H(3,𝔽3)C2211yescharacteristic
11C2×C3⋊#272C2211yescharacteristic
12C2×D3×H(3,𝔽3)C2211yescharacteristic
13C3⋊#1084C2211yescharacteristic
14C3×#1084C2211yescharacteristic
15C3×#1084C2211yescharacteristic
16Dic3×H(3,𝔽3)C2211yescharacteristic
17C3×#638—34no—
18C32×#67D314yescomplemented
19C2×C32×Dic3—34no—
20C3×#627—34no—
21C3×#626—34no—
22C3×#624—34no—
23C3×D3×Dic3—34no—
24D4×H(3,𝔽3)—31no—
25#6235—31no—
26#6235—31no—
27#6235—64no—
28C2×#1084—31no—
29C2×#1084D311yescharacteristic, complemented
30C4×#272—31no—
31#6229—31no—
32C2×C3×H(3,𝔽3)—21no—
33C2×C3×H(3,𝔽3)C2311yescharacteristic
34C3⋊#272—21no—
35D3×H(3,𝔽3)—21no—
36C3×#237—94no—
37C3×#235—91no—
38#3804—34no—
39C22×C33C2×D314yes—
40C2×C3×C3⋊D3—34no—
41C2×C32×D3C2×D314yes—
42C3×C3⋊Dic3—34no—
43C32×Dic3—34no—
44C32×Dic3—34no—
45C32×Dic3C2×D314yes—
46C22×H(3,𝔽3)—24no—
47C22×H(3,𝔽3)—31no—
48C22×H(3,𝔽3)C2×D311yescharacteristic
49C2×#272—31no—
50C2×#272—64no—
51C4×H(3,𝔽3)—31no—
52#1084—31no—
53#1084—64no—
54#1084C2×D311yescharacteristic
55#1084C2×D311yescharacteristic
56C3×H(3,𝔽3)#4111yescharacteristic, complemented, Sylow
57C32×D4—34no—
58C2×C32×C4—91no—
59#638—34no—
60C3×#67—34no—
61C3×#67—94no—
62C3×#67—94no—
63C3×#67—91no—
64C3×#67—184no—
65C3×#67C3⋊D311yescharacteristic
66C2×C3×Dic3—34no—
67C2×C3×Dic3—34no—
68C2×C3×Dic3—64no—
69C2×C3×Dic3—94no—
70C3×D12—91no—
71C3×C4×D3—94no—
72C3×C4×D3—91no—
73C3×C4×D3—91no—
74C3×Dic6—94no—
75C3×Dic6—91no—
76#627—34no—
77#626—34no—
78#624—34no—
79D3×Dic3—34no—
80C2×C33—24no—
81C2×C33C22×D314yes—
82C3×C3⋊D3—64no—
83C32×D3—24no—
84C2×H(3,𝔽3)—21no—
85C2×H(3,𝔽3)—24no—
86C2×H(3,𝔽3)—44no—
87C2×H(3,𝔽3)—61no—
88C2×H(3,𝔽3)C22×D311yescharacteristic
89#272—61no—
90#272—124no—
91C3×#41—271no—
92#237—94no—
93#235—91no—
94C22×C32—24no—
95C22×C32—34no—
96C22×C32—34no—
97C22×C32—64no—
98C22×C32D3214yes—
99C22×C32C2×C3⋊D311yescharacteristic
100C2×C3⋊D3—34no—
101C2×C3×D3—34no—
102C2×C3×D3—94no—
103C2×C3×D3—91no—
104C2×C3×D3—184no—
105C2×C3×D3C2×C3⋊D311yescharacteristic
106C32×C4—34no—
107C32×C4—91no—
108C32×C4—91no—
109C3⋊Dic3—34no—
110C3×Dic3—34no—
111C3×Dic3—34no—
112C3×Dic3—34no—
113C3×Dic3—34no—
114C3×Dic3—34no—
115C3×Dic3—64no—
116C3×Dic3—64no—
117C3×Dic3—94no—
118C3×Dic3—91no—
119C3×Dic3—184no—
120C3×Dic3C2×C3⋊D311yescharacteristic
121C33#23714yescomplemented
122H(3,𝔽3)—24no—
123H(3,𝔽3)#23511yescharacteristic, complemented
124C3×Q8—271no—
125C3×D4—31no—
126C3×D4—94no—
127C3×D4—271no—
128C3×D4—271no—
129C2×C3×C4—91no—
130C2×C3×C4—91no—
131C2×C3×C4—181no—
132C2×C3×C4—271no—
133C2×C3×C4—271no—
134#67—94no—
135#67—94no—
136#67—91no—
137#67—184no—
138#67#27211yescharacteristic, complemented
139C2×Dic3—34no—
140C2×Dic3—94no—
141D12—91no—
142C4×D3—94no—
143C4×D3—91no—
144C4×D3—91no—
145Dic6—94no—
146Dic6—91no—
147C2×C32—21no—
148C2×C32—24no—
149C2×C32—24no—
150C2×C32—34no—
151C2×C32—44no—
152C2×C32—64no—
153C2×C32—64no—
154C2×C32—68no—
155C2×C32—64no—
156C2×C32C2×D3214yes—
157C2×C32C22×C3⋊D311yescharacteristic
158C3⋊D3—64no—
159C3×D3—21no—
160C3×D3—64no—
161C3×D3—184no—
162C3×D3—181no—
163C3×D3—364no—
164#41—271noSylow
165C22×C3—21no—
166C22×C3—31no—
167C22×C3—34no—
168C22×C3—64no—
169C22×C3—94no—
170C22×C3—271no—
171C22×C3C2×#27211yescharacteristic
172C22×C3D3×C3⋊D311yescharacteristic
173C2×D3—94no—
174C2×D3—91no—
175C2×D3—184no—
176C2×D3C2×#27211yescharacteristic
177C3×C4—31no—
178C3×C4—94no—
179C3×C4—91no—
180C3×C4—91no—
181C3×C4—91no—
182C3×C4—91no—
183C3×C4—181no—
184C3×C4—181no—
185C3×C4—271no—
186Dic3—34no—
187Dic3—34no—
188Dic3—94no—
189Dic3—91no—
190Dic3—184no—
191Dic3C2×#27211yescharacteristic
192C32—24no—
193C32—34no—
194C32—64no—
195C32#380414yescomplemented
196C32#383011yescharacteristic
197Q8—271no—
198D4—31no—
199D4—271no—
200D4—271no—
201C2×C4—91no—
202C2×C4—271no—
203C2×C4—271no—
204C2×C3—22no—
205C2×C3—21no—
206C2×C3—21no—
207C2×C3—21no—
208C2×C3—34no—
209C2×C3—64no—
210C2×C3—61no—
211C2×C3—64no—
212C2×C3—124no—
213C2×C3—184no—
214C2×C3—541no—
215C2×C3C22×#27211yescharacteristic
216C2×C3C2×D3×C3⋊D311yescharacteristic
217D3—21no—
218D3—184no—
219D3—181no—
220D3—364no—
221C22—31no—
222C22—271no—
223C22D3×#27211yescharacteristic
224C4—31no—
225C4—91no—
226C4—91no—
227C4—271no—
228C3—21no—
229C3—34no—
230C3—64no—
231C3#8874411yescharacteristic, complemented
232C3#8902811yescharacteristic
233C2—21no—
234C2—61no—
235C2—541no—
236C2C2×D3×#27211yescharacteristic
subgroupG/H
center, FrattiniC2×C3C2×D3×C3⋊D3
commutator, cosocleC2×C3×H(3,𝔽3)C23
FittingC22×C3×H(3,𝔽3)C22
socleC2×C32C22×C3⋊D3
hypercenterC22×C3D3×C3⋊D3
hypocenterC3×H(3,𝔽3)#41
LCSG ▷ C2×C3×H(3,𝔽3) ▷ C3×H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C22×C3

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #10619994C2semidirect
2. C2×C3×#1084C2semidirect
3. #67×H(3,𝔽3)C2semidirect
4. #10619894C2semidirect
5. #10619892C2semidirect
6. #10619890C2semidirect
7. D3×#1084C2semidirect
18. C32×#67D3semidirect
29. C2×#1084D3semidirect
56. C3×H(3,𝔽3)#41semidirect
121. C33#237semidirect
123. H(3,𝔽3)#235semidirect
138. #67#272semidirect
195. C32#3804semidirect
231. C3#88744semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#106199942
C2×C3×#10842
#67×H(3,𝔽3)2
#106198942
#106198922
#106198902
D3×#10842
C3×#38043
#887443